The dialectic of math education

Thursday, June 23, 2011 — Adam Kotsko

Neoliberal educational ideology would have it that education is finally about job skills. Many have lamented this narrow focus as an impoverished view of what education should do and have argued in favor of a renewed emphasis on the humanities (in the interest of more holistic formation of character, training up in the duties of critical citizenship, etc.). I am very sympathetic with all those views. I would like to suggest, however, that they do not go far enough, because they accept the notion that it’s possible for formal education to be directed purely at job skills.

Yet there’s always something excessive in formal education, something that cannot be captured in a pure utilitarian calculation — that’s what makes it “education” rather than simple “training.” Furthermore, that excessive element corresponds to society’s own self-image. In what it forces kids to learn, over and above any straightforward utility, a society is telling a story about itself and its aspirations.

In a previous regime of education, the excessive element was classical education and particularly the study of the Latin language, a practice that continued (among elites) long past the point of Latin’s actual use as a common European language. It’s not difficult to understand the message behind this insistence on the importance of a dead language — it expressed a sense of connection to a broader Western heritage as well as to the task of universal empire-building. Presumably all those goals could be met by reading the works in question in translation, or else distilling their insights and strategies into contemporary works, making learning the actual Latin language excessive — yet that excessiveness is precisely the point.

In the contemporary U.S., the equivalent of the Latin language is surely mathematics. Indeed, math is frequently evoked as one of the first types of skills that one needs for today’s job market. Thinking back, however, I learned a lot of things in math class that were excessive. I’ve never needed to use algebra or geometry to any serious degree, let alone calculus.

Obviously I’m not in a field that demands a lot of calculation, but I know many engineers who report that their hard-won knowledge of differential equations has proven useless for their chosen vocation as well. Even basic math skills such as addition and subtraction are probably excessive for most practical purposes in the era of the cheap calculator — if you have good enough estimation skills to know when you’ve made an obvious error punching in the numbers, that’s probably going to be perfectly adequate 90% of the time. Similarly, I’d imagine that even in very advanced fields of applied mathematics, such as creating actuarial tables, etc., it’s more important to learn how to use the relevant software tools than to be able to do all the pure mathematics by hand.

What message was American society sending itself when it forced me to memorize my multiplication tables and laboriously solve quadratic equations? As with classical Latin, the stated rationale is to enable a kind of mastery, but the difference between Latin and math is that math is (at least in the common sense view) ahistorical and universal — meaning it is bound by neither ideology nor culture. The kind of mastery that math gives us is a non-ideological, pure mastery.

Hence the durability of the emphasis on math after the fall of the Soviet Union (whose engineering successes prompted the focus on math in the first place) — the Cold War was not perceived as a battle between competing ideologies, but a battle between the natural order of things (liberal democratic capitalism) and ideology as such (the Soviet Union). Now that ideology has been defeated after its failed attempt to instrumentalize the objective power of math for its non-objective ends, all nations can compete on the level playing field of math.

The seemingly indisputable nature of math has made our reliance on it much more dangerous than the previous reliance on Latin. While nationalistic pride in the mother tongue ultimately displaced Latin, there is apparently no counterweight to math, no counterargument.

Our trust in math became toxic, as the finance industry clothed its gambling more and more in the language of advanced math — and syphoned off as many advanced mathematicians as possible, diverting them from the practical tasks that math was supposed to serve. Surely the big banks must really be allocating risk in the best possible way: look at all that math! Surely derivatives must be self-regulating: did you see those fucking equations? These guys must know what they’re doing, because it’s all based in the very standard of knowledge itself! They must deserve the power they have, because it’s all based in the source of all objective, non-ideological power!

All of this, of course, makes about as much sense as it would have for all the nations of Europe to do whatever the pope says because he’s such a master of Latin — or worse, to continue to do whatever the pope says even after his advice caused widespread disaster and suffering. Yet that’s exactly what we’re doing. Math still reigns supreme, as we continue to tell ourselves the story that’s making us stupider by the day.

In this perspective, the perversely ineffective mode of American math education, which focuses on “skills” without providing any conceptual background to let students know why they’re learning to shift numbers and letters around according to these rules, makes perfect sense. Just as American-style foreign language education exists solely to convince the average American that learning to speak a foreign language is impossible, so also is American-style math education designed to convince the majority of students that math is too hard for them. It’s a kind of inoculation, exposing them to just enough math to convince them not to ask questions about it — perhaps not unlike the use of Latin in legal language.

All of this does provide valuable job skills, albeit indirectly: skills in trusting the authority of elites and in accepting any mathematically-legitimated decision as a fact of nature (“they have to make money somehow!”; “there just isn’t room in the budget!”). And actually doing well on the math itself really does position one to be among the best-renumerated servants of the elites.

But I’m pretty sure that the elites themselves mostly aren’t very good at algebra — they’re the ones who somehow stumbled on enough “critical thinking skills” to know that they can always hire someone else to do that for them.

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14 Responses to “The dialectic of math education”

Had I done more math in high school, I likely would have received higher grades in my required statistics courses. Is this what they mean by job skills, especially given that I don’t actually use statistics or math (beyond “Stop whining; it is only thirty pages for next week! You’ll survive!”) in my work?

In my required math course for non-math majors, one student asked the age-old question “What do you even use this stuff for?” The teacher assistant answered “To pass this course, which you need for your degree.” I’m sympathetic to your post here but I kind of feel that having knowledge of statistical math and the like (since they’re used so frequently, especially in the media) would be awesome to have. I’ll have to get to work on it.

Another solution would be for the media to report statistics more responsibly — although just to clarify, I’m not against math per se, nor even necessarily against requiring students to learn it. I’m just against the ideology in which math education is entangled.

It seems to me that most of what people are talking when they talk about math(s) in these contexts is what I would call “reckoning”. It useful to know how to reckon, but also useful to know the limits of the particular reckoning techniques employed in different applications (so as not to misapply statistics, for example). The “knowing the limits” part is much more like what I would actually call “maths” – hegemonic dominance of which would be, in my view, an unprecedented wonder.

Random thought about the banks-I once heard or read somewhere that some banks and investing firms will employ extremely advanced mathematical equations in their records in order to make it difficult for somebody to look and in and check over what they are in fact doing. Sneeky.

I think the emphasis of your argument is just a little bit off. The one phrase that I think really captured the financial failure is “mistaking beauty for truth” that appears in Krugman’s seminal essay in the NYT Magazine, “How Did Economists Get It So Wrong?” (http://nyti.ms/1f5SpE). Krugman’s principle argument is that economists and Wall Streeters mistook mathematically dense arguments for truth, and it so happens that neoliberal (or in his words, “neoclassical”) economic theory is more amenable to analytical mathematical techniques because of its assumption that markets work and people are 100% rational.

So, does this mean that excessive faith in math caused the crisis? Close, but not quite IMO. When you make the correct assumptions and you do your math right, then you’ll get the correct answer. Math isn’t wrong, it doesn’t lead one astray. As I see it, the economic crisis happened because of our excessive desire for simple answers. Greed simply served to further fan the flames of this desire. The desire for simplicity, sometimes mistaken for Occam’s Razor, is very much present in my field of study, physics. For a long time, physicists were unwilling to look into the complex behavior of nonlinear systems because they were notoriously difficult to make analytical progress on. It wasn’t until the advent of relatively cheap computers and numerical solving techniques that they made progress in such fields and paved the way for chaos and complexity theory (“analytical” refers to, e.g., solving the equation x+5=2x and getting the solution x=5 like you learned in school, whereas a numerical/computer technique would be to plot the lines x+5 and 2x on a Cartesian plane with a computer and seeing that they intersect above the x=5 ticmark of the x-axis). Still, computer solutions do not offer the ease and versatility that analytical solutions can and chaos theory doesn’t even really offer specific solutions to a problem at all (cf. the “butterfly effect”). In short, analytical techniques offer easy and versatile solutions to problems and are therefore enormously useful, but you typically can only use analytical techniques if you make simplifying assumptions about the system you are studying, and these assumptions are oftentimes wrong. Computer and numerical economic models that attempt to account for the fact that humans aren’t 100% rational and that markets fail are perhaps ignored because they won’t give simple answers that one can make money on. To sum up with a H. L. Mencken quote that Krugman uses to conclude his essay: “There is always an easy solution to every human problem — neat, plausible and wrong.”

Well, there’s an additional problem: modern mathematics is placed on a different philosophic foundation than scholastic math (meaning math in the Middle Ages), and ancient Greek math (the Romans did very little work in mathematics or science in general). There are philosophic assumptions buried within mathematical theory. You can only get a real grasp on this issue by examining ancient scientific texts – most of which have not been translated. Some of the work of an epochal figure like Albertus Magnus still remains untranslated into English, for instance. Lack of knowledge of the classical languages actually does pose a barrier to popular enlightenment right now.

That math itself has changed indicates the possibility that math itself (or, more correctly, our understanding of math) is neither objective nor non-ideological.

Fundamentally, the problem is even worse than you portray: previous societies usually based themselves on some understanding of their gods (or at least propagandized their populations to think so). But strangely, divine revelation is more democratic than high status within modern science: the gods could talk to anybody (in theory) while getting high scientific status is limited to very few individuals. Effectively, for most people, the faith in god (or gods) has been replaced by a faith in science. And that faith in science isn’t itself necessarily more rational than the old faiths in the gods – both are irrational, but very few people examine the foundations of science, while many more (seem to) examine the foundations of their gods.

That means that current society might easily be more susceptible to being convinced by scientific charlatans than earlier societies were susceptible to being convinced by religious charlatans. Just look at how many wildly different economic ideologies have motivated governments just in the past century. And the partisans of each ideology believed that their ideology was supported by science (indeed, almost all saw their economic ideologies as part of science simply and not as something supported by science) and were willing to do almost anything to impose it on a worldwide basis.

[…] Adam Kotsko on education and the excessiveness of math: [T]here’s always something excessive in formal education, something that cannot be captured in a pure utilitarian calculation — that’s what makes it “education” rather than simple “training.” Furthermore, that excessive element corresponds to society’s own self-image. In what it forces kids to learn, over and above any straightforward utility, a society is telling a story about itself and its aspirations. […]

Excellent thoughts, Adam. As a former language teacher, I’m intrigued by your observation that American-style foreign language instruction is designed to reinforce Americans’ sense of the impossibility of speaking a foreign language, our sense of Americans as “bad at languages”, etc. If the mystique of math as elite expertise serves as a fig leaf for the reign of the ‘quants’ who run global capitalism, I wonder what ideological agenda might be served by the mystification of US foreign language education. Do you see it operating in a similar way to math, as a deliberately obfuscation of knowledge and know-how meant to act as ‘bitter pill’ to signal to the masses to leave well enough alone, a message of “Experts only past this point” (where “experts” means the well-positioned darlings of the technocratic knowledge class under neoliberalism, etc.)?
I have my ideas about how this might tie into that ‘exorbitant privilege’ Americans enjoy in the linguistic/cultural sphere by virtue of English’s role as lingua franca, paralleling the role the currency plays in the world economy, but your comment really got me thinking.

I’m sympathetic towards your criticism of the elite “quants” employed by Wall Street firms. It is unfortunate that some of the best minds out there are co-opted into thwarting regulators.

At the same time, I think your critique of mathematics is overblown. The amount of math the average American is expected to know is trivial at best. Algebra, Geometry, maybe Trig. None of which should be very difficult to anyone who applies some sustained effort. Your comparison of math with foreign languages is correct in this regard. Both take years of practice and daily exercise to become proficient.

I think the real horror is that American students have demonstrated themselves unable to excel in Math while doing well in other fields. Surely, teachers and parents have some responsibility in this, but there’d be no need to emphasize math at the expense of the humanities or the arts if students could demonstrate a reasonable level of proficiency. Simply expecting a student to solve a “quadratic equation” (THE HORROR! YOU MIGHT HAVE TO FACTOR!) shouldn’t be the crucible it’s made out to be.

Jamie, Where is my critique of mathematics as such? I enjoyed math as a student and wish I’d taken more, just for my own satisfaction. If people are reading this post and thinking that I’m saying math sucks, they’re really not reading carefully.

Will, I think that the inculcation of helplessness works differently in each case. In terms of foreign languages, I think that Americans’ learned helplessness comes to reinforce the presumption that “everyone else” must learn English — which then creates a vicious cycle where, for instance, foreigners switch immediately to English when Americans try to practice speaking with them, because they (pragmatically!) view practicing their English as more of a priority. More generally, it seems to reinforce American exceptionalism and the sense that the way we do things is just immediately “normal,” while everyone else is foreign or weird in some way. It probably also helps out with the ideology of assimilation — there’s a lot more incentive to learn English when basically no one is even in a position to cut you any slack.